*> \brief \b DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAEIN + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, * LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO ) * * .. Scalar Arguments .. * LOGICAL NOINIT, RIGHTV * INTEGER INFO, LDB, LDH, N * DOUBLE PRECISION BIGNUM, EPS3, SMLNUM, WI, WR * .. * .. Array Arguments .. * DOUBLE PRECISION B( LDB, * ), H( LDH, * ), VI( * ), VR( * ), * \$ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAEIN uses inverse iteration to find a right or left eigenvector *> corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg *> matrix H. *> \endverbatim * * Arguments: * ========== * *> \param[in] RIGHTV *> \verbatim *> RIGHTV is LOGICAL *> = .TRUE. : compute right eigenvector; *> = .FALSE.: compute left eigenvector. *> \endverbatim *> *> \param[in] NOINIT *> \verbatim *> NOINIT is LOGICAL *> = .TRUE. : no initial vector supplied in (VR,VI). *> = .FALSE.: initial vector supplied in (VR,VI). *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix H. N >= 0. *> \endverbatim *> *> \param[in] H *> \verbatim *> H is DOUBLE PRECISION array, dimension (LDH,N) *> The upper Hessenberg matrix H. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> The leading dimension of the array H. LDH >= max(1,N). *> \endverbatim *> *> \param[in] WR *> \verbatim *> WR is DOUBLE PRECISION *> \endverbatim *> *> \param[in] WI *> \verbatim *> WI is DOUBLE PRECISION *> The real and imaginary parts of the eigenvalue of H whose *> corresponding right or left eigenvector is to be computed. *> \endverbatim *> *> \param[in,out] VR *> \verbatim *> VR is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[in,out] VI *> \verbatim *> VI is DOUBLE PRECISION array, dimension (N) *> On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain *> a real starting vector for inverse iteration using the real *> eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI *> must contain the real and imaginary parts of a complex *> starting vector for inverse iteration using the complex *> eigenvalue (WR,WI); otherwise VR and VI need not be set. *> On exit, if WI = 0.0 (real eigenvalue), VR contains the *> computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), *> VR and VI contain the real and imaginary parts of the *> computed complex eigenvector. The eigenvector is normalized *> so that the component of largest magnitude has magnitude 1; *> here the magnitude of a complex number (x,y) is taken to be *> |x| + |y|. *> VI is not referenced if WI = 0.0. *> \endverbatim *> *> \param[out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB,N) *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= N+1. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[in] EPS3 *> \verbatim *> EPS3 is DOUBLE PRECISION *> A small machine-dependent value which is used to perturb *> close eigenvalues, and to replace zero pivots. *> \endverbatim *> *> \param[in] SMLNUM *> \verbatim *> SMLNUM is DOUBLE PRECISION *> A machine-dependent value close to the underflow threshold. *> \endverbatim *> *> \param[in] BIGNUM *> \verbatim *> BIGNUM is DOUBLE PRECISION *> A machine-dependent value close to the overflow threshold. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> = 1: inverse iteration did not converge; VR is set to the *> last iterate, and so is VI if WI.ne.0.0. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleOTHERauxiliary * * ===================================================================== SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, \$ LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. LOGICAL NOINIT, RIGHTV INTEGER INFO, LDB, LDH, N DOUBLE PRECISION BIGNUM, EPS3, SMLNUM, WI, WR * .. * .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), H( LDH, * ), VI( * ), VR( * ), \$ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TENTH PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TENTH = 1.0D-1 ) * .. * .. Local Scalars .. CHARACTER NORMIN, TRANS INTEGER I, I1, I2, I3, IERR, ITS, J DOUBLE PRECISION ABSBII, ABSBJJ, EI, EJ, GROWTO, NORM, NRMSML, \$ REC, ROOTN, SCALE, TEMP, VCRIT, VMAX, VNORM, W, \$ W1, X, XI, XR, Y * .. * .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DASUM, DLAPY2, DNRM2 EXTERNAL IDAMAX, DASUM, DLAPY2, DNRM2 * .. * .. External Subroutines .. EXTERNAL DLADIV, DLATRS, DSCAL * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, SQRT * .. * .. Executable Statements .. * INFO = 0 * * GROWTO is the threshold used in the acceptance test for an * eigenvector. * ROOTN = SQRT( DBLE( N ) ) GROWTO = TENTH / ROOTN NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM * * Form B = H - (WR,WI)*I (except that the subdiagonal elements and * the imaginary parts of the diagonal elements are not stored). * DO 20 J = 1, N DO 10 I = 1, J - 1 B( I, J ) = H( I, J ) 10 CONTINUE B( J, J ) = H( J, J ) - WR 20 CONTINUE * IF( WI.EQ.ZERO ) THEN * * Real eigenvalue. * IF( NOINIT ) THEN * * Set initial vector. * DO 30 I = 1, N VR( I ) = EPS3 30 CONTINUE ELSE * * Scale supplied initial vector. * VNORM = DNRM2( N, VR, 1 ) CALL DSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), VR, \$ 1 ) END IF * IF( RIGHTV ) THEN * * LU decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * DO 60 I = 1, N - 1 EI = H( I+1, I ) IF( ABS( B( I, I ) ).LT.ABS( EI ) ) THEN * * Interchange rows and eliminate. * X = B( I, I ) / EI B( I, I ) = EI DO 40 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 40 CONTINUE ELSE * * Eliminate without interchange. * IF( B( I, I ).EQ.ZERO ) \$ B( I, I ) = EPS3 X = EI / B( I, I ) IF( X.NE.ZERO ) THEN DO 50 J = I + 1, N B( I+1, J ) = B( I+1, J ) - X*B( I, J ) 50 CONTINUE END IF END IF 60 CONTINUE IF( B( N, N ).EQ.ZERO ) \$ B( N, N ) = EPS3 * TRANS = 'N' * ELSE * * UL decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * DO 90 J = N, 2, -1 EJ = H( J, J-1 ) IF( ABS( B( J, J ) ).LT.ABS( EJ ) ) THEN * * Interchange columns and eliminate. * X = B( J, J ) / EJ B( J, J ) = EJ DO 70 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 70 CONTINUE ELSE * * Eliminate without interchange. * IF( B( J, J ).EQ.ZERO ) \$ B( J, J ) = EPS3 X = EJ / B( J, J ) IF( X.NE.ZERO ) THEN DO 80 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - X*B( I, J ) 80 CONTINUE END IF END IF 90 CONTINUE IF( B( 1, 1 ).EQ.ZERO ) \$ B( 1, 1 ) = EPS3 * TRANS = 'T' * END IF * NORMIN = 'N' DO 110 ITS = 1, N * * Solve U*x = scale*v for a right eigenvector * or U**T*x = scale*v for a left eigenvector, * overwriting x on v. * CALL DLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, \$ VR, SCALE, WORK, IERR ) NORMIN = 'Y' * * Test for sufficient growth in the norm of v. * VNORM = DASUM( N, VR, 1 ) IF( VNORM.GE.GROWTO*SCALE ) \$ GO TO 120 * * Choose new orthogonal starting vector and try again. * TEMP = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 DO 100 I = 2, N VR( I ) = TEMP 100 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 110 CONTINUE * * Failure to find eigenvector in N iterations. * INFO = 1 * 120 CONTINUE * * Normalize eigenvector. * I = IDAMAX( N, VR, 1 ) CALL DSCAL( N, ONE / ABS( VR( I ) ), VR, 1 ) ELSE * * Complex eigenvalue. * IF( NOINIT ) THEN * * Set initial vector. * DO 130 I = 1, N VR( I ) = EPS3 VI( I ) = ZERO 130 CONTINUE ELSE * * Scale supplied initial vector. * NORM = DLAPY2( DNRM2( N, VR, 1 ), DNRM2( N, VI, 1 ) ) REC = ( EPS3*ROOTN ) / MAX( NORM, NRMSML ) CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) END IF * IF( RIGHTV ) THEN * * LU decomposition with partial pivoting of B, replacing zero * pivots by EPS3. * * The imaginary part of the (i,j)-th element of U is stored in * B(j+1,i). * B( 2, 1 ) = -WI DO 140 I = 2, N B( I+1, 1 ) = ZERO 140 CONTINUE * DO 170 I = 1, N - 1 ABSBII = DLAPY2( B( I, I ), B( I+1, I ) ) EI = H( I+1, I ) IF( ABSBII.LT.ABS( EI ) ) THEN * * Interchange rows and eliminate. * XR = B( I, I ) / EI XI = B( I+1, I ) / EI B( I, I ) = EI B( I+1, I ) = ZERO DO 150 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - XR*TEMP B( J+1, I+1 ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 150 CONTINUE B( I+2, I ) = -WI B( I+1, I+1 ) = B( I+1, I+1 ) - XI*WI B( I+2, I+1 ) = B( I+2, I+1 ) + XR*WI ELSE * * Eliminate without interchanging rows. * IF( ABSBII.EQ.ZERO ) THEN B( I, I ) = EPS3 B( I+1, I ) = ZERO ABSBII = EPS3 END IF EI = ( EI / ABSBII ) / ABSBII XR = B( I, I )*EI XI = -B( I+1, I )*EI DO 160 J = I + 1, N B( I+1, J ) = B( I+1, J ) - XR*B( I, J ) + \$ XI*B( J+1, I ) B( J+1, I+1 ) = -XR*B( J+1, I ) - XI*B( I, J ) 160 CONTINUE B( I+2, I+1 ) = B( I+2, I+1 ) - WI END IF * * Compute 1-norm of offdiagonal elements of i-th row. * WORK( I ) = DASUM( N-I, B( I, I+1 ), LDB ) + \$ DASUM( N-I, B( I+2, I ), 1 ) 170 CONTINUE IF( B( N, N ).EQ.ZERO .AND. B( N+1, N ).EQ.ZERO ) \$ B( N, N ) = EPS3 WORK( N ) = ZERO * I1 = N I2 = 1 I3 = -1 ELSE * * UL decomposition with partial pivoting of conjg(B), * replacing zero pivots by EPS3. * * The imaginary part of the (i,j)-th element of U is stored in * B(j+1,i). * B( N+1, N ) = WI DO 180 J = 1, N - 1 B( N+1, J ) = ZERO 180 CONTINUE * DO 210 J = N, 2, -1 EJ = H( J, J-1 ) ABSBJJ = DLAPY2( B( J, J ), B( J+1, J ) ) IF( ABSBJJ.LT.ABS( EJ ) ) THEN * * Interchange columns and eliminate * XR = B( J, J ) / EJ XI = B( J+1, J ) / EJ B( J, J ) = EJ B( J+1, J ) = ZERO DO 190 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - XR*TEMP B( J, I ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 190 CONTINUE B( J+1, J-1 ) = WI B( J-1, J-1 ) = B( J-1, J-1 ) + XI*WI B( J, J-1 ) = B( J, J-1 ) - XR*WI ELSE * * Eliminate without interchange. * IF( ABSBJJ.EQ.ZERO ) THEN B( J, J ) = EPS3 B( J+1, J ) = ZERO ABSBJJ = EPS3 END IF EJ = ( EJ / ABSBJJ ) / ABSBJJ XR = B( J, J )*EJ XI = -B( J+1, J )*EJ DO 200 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - XR*B( I, J ) + \$ XI*B( J+1, I ) B( J, I ) = -XR*B( J+1, I ) - XI*B( I, J ) 200 CONTINUE B( J, J-1 ) = B( J, J-1 ) + WI END IF * * Compute 1-norm of offdiagonal elements of j-th column. * WORK( J ) = DASUM( J-1, B( 1, J ), 1 ) + \$ DASUM( J-1, B( J+1, 1 ), LDB ) 210 CONTINUE IF( B( 1, 1 ).EQ.ZERO .AND. B( 2, 1 ).EQ.ZERO ) \$ B( 1, 1 ) = EPS3 WORK( 1 ) = ZERO * I1 = 1 I2 = N I3 = 1 END IF * DO 270 ITS = 1, N SCALE = ONE VMAX = ONE VCRIT = BIGNUM * * Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, * or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector, * overwriting (xr,xi) on (vr,vi). * DO 250 I = I1, I2, I3 * IF( WORK( I ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) SCALE = SCALE*REC VMAX = ONE VCRIT = BIGNUM END IF * XR = VR( I ) XI = VI( I ) IF( RIGHTV ) THEN DO 220 J = I + 1, N XR = XR - B( I, J )*VR( J ) + B( J+1, I )*VI( J ) XI = XI - B( I, J )*VI( J ) - B( J+1, I )*VR( J ) 220 CONTINUE ELSE DO 230 J = 1, I - 1 XR = XR - B( J, I )*VR( J ) + B( I+1, J )*VI( J ) XI = XI - B( J, I )*VI( J ) - B( I+1, J )*VR( J ) 230 CONTINUE END IF * W = ABS( B( I, I ) ) + ABS( B( I+1, I ) ) IF( W.GT.SMLNUM ) THEN IF( W.LT.ONE ) THEN W1 = ABS( XR ) + ABS( XI ) IF( W1.GT.W*BIGNUM ) THEN REC = ONE / W1 CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) XR = VR( I ) XI = VI( I ) SCALE = SCALE*REC VMAX = VMAX*REC END IF END IF * * Divide by diagonal element of B. * CALL DLADIV( XR, XI, B( I, I ), B( I+1, I ), VR( I ), \$ VI( I ) ) VMAX = MAX( ABS( VR( I ) )+ABS( VI( I ) ), VMAX ) VCRIT = BIGNUM / VMAX ELSE DO 240 J = 1, N VR( J ) = ZERO VI( J ) = ZERO 240 CONTINUE VR( I ) = ONE VI( I ) = ONE SCALE = ZERO VMAX = ONE VCRIT = BIGNUM END IF 250 CONTINUE * * Test for sufficient growth in the norm of (VR,VI). * VNORM = DASUM( N, VR, 1 ) + DASUM( N, VI, 1 ) IF( VNORM.GE.GROWTO*SCALE ) \$ GO TO 280 * * Choose a new orthogonal starting vector and try again. * Y = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 VI( 1 ) = ZERO * DO 260 I = 2, N VR( I ) = Y VI( I ) = ZERO 260 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 270 CONTINUE * * Failure to find eigenvector in N iterations * INFO = 1 * 280 CONTINUE * * Normalize eigenvector. * VNORM = ZERO DO 290 I = 1, N VNORM = MAX( VNORM, ABS( VR( I ) )+ABS( VI( I ) ) ) 290 CONTINUE CALL DSCAL( N, ONE / VNORM, VR, 1 ) CALL DSCAL( N, ONE / VNORM, VI, 1 ) * END IF * RETURN * * End of DLAEIN * END